A study of minimizing sequences. (English) Zbl 0553.49017

Polak, E.; Wardi, Y. Y.

SIAM J. Control Optimization 22, 599-609 (1984).

The authors consider the problem P: \(\inf\{f(x): x\in X\}\) where X is a subset of a normed space X and \(f: X\to {\mathbb{R}}\) is bounded from below on X and uniformly continuous on bounded sets. A sequence \(\{x_ i\}^{\infty}_{i=0}\subset X\) is called eventually feasible for P if \(\lim_{i\to \infty}\inf \{\| x_ i-x\| | x\in X\}=0\). A bounded eventually feasible sequence \(\{\hat x_ i\}\) is called locally minimizing for P if there exists a \(\rho >0\) such that for all eventually feasible sequences \(\{x_ i\}\) satisfying \(\limsup_{i\to \infty}\| \hat x_ i-x_ i\| \leq \rho\) we have \(\limsup_{i\in K, i\to \infty}f(\hat x_ i)\leq\limsup_{i\in K,i\to \infty}f(x_ i)\) for all infinite subsets \(K\subset \{0,1,2,...\}\). Next, the authors define \(\tilde X\) to be the class of all infinite sequences \(\{x_ i\}\subset X\), and the vector space \(X^ s\) to be \(\tilde X/\sim\) where \(\sim\) is the equivalence relation defined as follows: \(\{x_ i\}\sim \{y_ i\}\) if and only if \(\lim_{i\to \infty}\| x_ i-y_ i\| =0\). Now, let \(z\in X^ s\) and let \(\{x_ i\}\) be any sequence in the equivalence class z. The following definitions are independent of the particular choice of this sequence: \(X^ s=\{z\in X^ s| \limsup_{i\to \infty}\| x_ i\| <\infty\) and \(\{x_ i\}\) is eventually feasible}, \(f^ s(z)=\limsup_{i\to \infty}f(x_ i)\). A point \(\hat z\in X^ s\) is called a local minimizer for the problem \(P^ s:\) \(\min \{f^ s(z)| z\in X^ s\}\) if \(\{\hat x_ i\}\) is a locally minimizing sequence for P, where \(\{\hat x_ i\}\) is any sequence in the equivalence class \(\hat z.\) The authors present some (first and second order) necessary and sufficient conditions for \(\hat z\) to be a local minimizer for \(P^ s\). It is shown that, for the case where f is differentiable, the standard optimality conditions for P lead directly to corresponding optimality conditions for \(P^ s\). The case where f is merely Lipschitz continuous is also considered and a first order necessary optimality condition (stated in terms of Clarke generalized gradients) is obtained. Finally, the authors consider some specific optimization problems with equality and inequality constraints and formulate optimality conditions for minimizing sequences for these problems.

Reviewer: M.Studniarski

MSC:

49K27	Optimality conditions for problems in abstract spaces
49M37	Numerical methods based on nonlinear programming
90C30	Nonlinear programming
65K05	Numerical mathematical programming methods
90C48	Programming in abstract spaces

Keywords:

eventually feasible sequence; local minimizer; minimizing sequence; optimality conditions